# Orbit-finite dimensional vector spaces, and orbit-finite systems of linear equations

Here is some naive questions that came to one’s mind when considering vector spaces where the dimension is not finite but orbit-finite: under a reasonable assumption on atoms , say -categoricity,

— is it true that every vector subspace of an orbit-finite dimensional vector space is orbit-finite dimensional?

— is it true that the set of solutions of an orbit-finite  system of linear equations is orbit-finite dimensional?

# Decidability results based on a WQO in homogeneous atoms

There are several decision problems for certain computation models with atoms that are decidable when atoms admit certain well quasi order (WQO),  and undecidable otherwise. We recall the problems and formulate few questions related to WQOs.

# The colored open question closed

A question in my recent post Automorphisms vs color-preserving automorphisms in Fraisse classes has been answered negatively by Mikołaj. The counterexample Fraisse class is the age of the structure , the rational numbers with the ternary ‘cyclic order’ relation defined by:

I am about to reformulate the question slightly, in order to make refutation harder;)

# Automorphisms vs color-preserving automorphisms in Fraisse classes

Here is a question closely related to a characterization of standard atoms (not to be confused with standard alphabets:). It asks whether a bound on the number of automorphisms is equivalent to  a bound on the numer of color-preserving automorphisms, in every Fraisse class of finite relational structures.

# Derived alphabets

We study a derivation quasi-order between between alphabets. Whenever is derived from and is non-standard then is non-standard too. We identify also a class of minimal non-standard alphabets wrt. the quasi-order.

# Least supports and the representation theorem

This post discusses equivalence of:

1. existence of the least supports

2. representation theorem